how to extrude an edge that stays connected with the face?
I assume that your wall is orthogonal to the x-y-plane. Then there are several options to extrude the floor without loosing "connection" e.g. lifting your wall up.

If your plane is level you can easily lock the extrude to a plane by Extruding and then (while the extrude is active) press shiftz this will lock the movement to the x-y-plane. In case you want the extrude to be rectangular you can press y during the extrude to lock it to the Y-Axis. (This can be a powerful technique if you are comfortable with the different coordinate system orientations in blender) Blender_Manual

The other method is, as pointed out in the comment, using vertex snapping with the target being a plane. https://www.blender.org/manual/editors/3dview/transform/transform_control/snap.html%3E